The Dog That Did Not Bark: A Defense of Return Predictability

Transcription

1 The Dog That Did Not Bark: A Defense of Return Predictability John H. Cochrane University of Chicago GSB and NBER If returns are not predictable, dividend growth must be predictable, to generate the observed variation in divided yields. I find that the absence of dividend growth predictability gives stronger evidence than does the presence of return predictability. Long-horizon return forecasts give the same strong evidence. These tests exploit the negative correlation of return forecasts with dividend-yield autocorrelation across samples, together with sensible upper bounds on dividend-yield autocorrelation, to deliver more powerful statistics. I reconcile my findings with the literature that finds poor power in long-horizon return forecasts, and with the literature that notes the poor out-of-sample R 2 of return-forecasting regressions. (JEL G12, G14, C22) Are stock returns predictable? Table 1 presents regressions of the real and excess value-weighted stock return on its dividend-price ratio, in annual data. In contrast to the simple random walk view, stock returns do seem predictable. Similar or stronger forecasts result from many variations of the variables and data sets. Economic significance The estimates in Table 1 have very large economic significance. The standard deviation of expected returns in the last column of Table 1 is about five percentage points, almost as large as the 7.7% level of the equity premium in this sample. The equity premium apparently varies over time by as much as its unconditional mean. The 4 7% R 2 do not look that impressive, but the R 2 rises with horizon, reaching values between 30 and 60%, depending on time period and estimation details, as emphasized by Fama and French (1988). The slope coefficient of over three in the top two rows means that when dividend yields rise one percentage point, prices rise another two percentage points on average, rather than declining one percentage point to offset the extra dividends and render returns unpredictable. Finally, the regressions of Table 1 imply that all variation in market price-dividend ratios corresponds to changes in expected excess I acknowledge research support from CRSP and from a NSF grant administered by the NBER. I thank Alan Bester, John Campbell, John Heaton, Lars Hansen, Anil Kashyap, Sydney Ludvigson, Lubos Pastor, Ivo Welch, and an anonymous referee for very helpful comments. Address correspondence to John H. Cochrane, Graduate School of Business, 5807 S. Woodlawn, Chicago IL 60637, , or The Author Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please doi: /rfs/hhm046 Advance Access publication September 22, 2007

2 The Review of Financial Studies / v 21 n Table 1 Forecasting regressions Regression b t R 2 (%) σ(bx)(%) R t+1 = a + b(d t /P t ) + ε t R t+1 Rt f = a + b(d t /P t ) + ε t D t+1 /D t = a + b(d t /P t ) + ε t r t+1 = a r + b r (d t p t ) + ε r t+1 d t+1 = a d + b d (d t p t ) + ε dp t R t+1 is the real return, deflated by the CPI, D t+1 /D t is real dividend growth, and D t /P t is the dividend-price ratio of the CRSP value-weighted portfolio. R f t+1 is the real return on 3-month Treasury-Bills. Small letters are logs of corresponding capital letters. Annual data, σ(bx) gives the standard deviation of the fitted value of the regression. returns risk premiums and none corresponds to news about future dividend growth. I present this calculation below. Statistical significance The statistical significance of the return forecast in Table 1 is marginal, however, with a t-statistic only a little above two. And the ink was hardly dry on the first studies 1 to run regressions like those of Table 1 before a large literature sprang up examining their econometric properties and questioning that statistical significance. The right-hand variable (dividend yield) is very persistent, and return shocks are negatively correlated with dividend-yield shocks. As a result, the return-forecast regression inherits the near-unit-root properties of the dividend yield. The coefficient is biased upward, and the t-statistic is biased toward rejection. Stambaugh (1986, 1999) derived the finite-sample distribution of the return-forecasting regression. In monthly regressions, Stambaugh found that in place of OLS p-values of 6% ( ) and 2% ( ), the correct p-values are 17 and 15%. The regressions are far from statistically significant at conventional levels. 2 Does this evidence mean return forecastability is dead? No, because there are more powerful tests, and these tests give stronger evidence against the null. First, we can examine dividend growth. In the regressions of Table 1, dividend growth is clearly not forecastable at all. In fact, the small point 1 Rozeff (1984), Shiller (1984), Keim and Stambaugh (1986), Campbell and Shiller (1988), and Fama and French (1988). 2 Precursors include Goetzmann and Jorion (1993) and Nelson and Kim (1993) who found the distribution of the return-forecasting coefficient by simulation and also did not reject the null. Mankiw and Shapiro (1986) show the bias point by simulation, though not applied to the dividend-yield/return case. Additional contributions include Kothari and Shanken (1997), Paye and Timmermann (2003), Torous, Valkanov, and Yan (2004), Campbell and Yogo (2006), and Ang and Bekaert (2007). 1534

3 A Defense of Return Predictability estimates have the wrong sign a high dividend yield means a low price, which should signal lower, not higher, future dividend growth. If both returns and dividend growth are unforecastable, then present value logic implies that the price/dividend ratio is constant, which it obviously is not. Alternatively, in the language of cointegration, since the dividend yield is stationary, one of dividend growth or price growth must be forecastable to bring the dividend yield back following a shock. We cannot just ask, Are returns forecastable? and Is dividend growth forecastable? We must ask, Which of dividend growth or returns is forecastable? (Or really, How much of each? ) A null hypothesis in which returns are not forecastable must also specify that dividend growth is forecastable, and the statistical evaluation of that null must also confront the lack of dividend-growth forecastability in the data. I set up such a null, and I evaluate the joint distribution of return and dividend-growth forecasting coefficients. I confirm that the return-forecasting coefficient, taken alone, is not significant: Under the unforecastable-return null, we see return forecast coefficients as large or larger thanthosein the dataabout20% ofthe timeand at-statistic as large as that seen in the data about 10% of the time. However, I find that the absence of dividend growth forecastability offers much more significant evidence against the null. The best overall number is a 1 2% probability value (last row of Table 5) dividend growth fails to be forecastable in only 1 2% of the samples generated under the null. The important evidence, as in Sherlock Holmes s famous case, is the dog that does not bark. 3 Second, we can examine the long-horizon return forecast implied by one-year regressions. It turns out to be most convenient to look at br lr b r / (1 ρφ) where φ is the dividend-yield autocorrelation, ρ 0.96 is a constant related to the typical level of the dividend yield, and b r is the return-forecast coefficient as defined in Table 1. The long horizon label applies because br lr is the implied coefficient of weighted long-horizon returns j=1 ρj 1 r t+j on dividend yields. The null hypothesis produces a long-horizon return regression coefficient br lr larger than its sample value only about 1 2% of the time, again delivering much stronger evidence against the null than the one-period return coefficient b r. Why are these tests more powerful? They exploit a feature of the data and a feature of the null hypothesis that the conventional b r test ignores. The feature of the data is that return shocks ε r and dividend-yield shocks ε dp are strongly and negatively correlated. Briefly, a price rise raises returns and lowers dividend yields. This correlation means that regression 3 Inspector Gregory: Is there any other point to which you would wish to draw my attention? Holmes: To the curious incident of the dog in the night-time. The dog did nothing in the night time. That was the curious incident. (From The Adventure of Silver Blaze by Arthur Conan Doyle.) 1535

4 The Review of Financial Studies / v 21 n estimates b r and φ are strongly and negatively correlated across samples. A large long-run coefficient br lr = b r /(1 ρφ) requires both a large b r and a large φ, so that the coefficients can build with horizon as they do in our data. But since b r and φ are negatively correlated, samples with unusually large b r tend to come with unusually low φ, so it is much harder for the null to generate large long-run coefficients. The dividend-growth test works the same way. The feature of the null is that we know something about the dividend-yield autocorrelation φ. The Wald test on b r uses no information about other parameters. It is the appropriate test of the null {b r = 0,φ = anything}. But we know φ cannot be too big. If φ>1/ρ 1.04, the present value relation explodes and the price-dividend ratio is infinite, which it also is obviously not. If φ 1.0, the dividend yield has a unit or larger root, meaning that its variance explodes with horizon. Economics, statistics, and common sense mean that if our null is to describe a coherent world, it should contain some upper bound on φ as well as b r = 0, something like { b r = 0, φ < φ }. A good test uses information on both ˆb r and ˆφ to evaluate such a null, drawing regions in {b r,φ} space around the null { b r = 0, φ < φ }, and exploiting the fact that under the null ˆb r should not be too big and ˆφ should not be too big. The test regions in {b r,φ} described by the long-run return coefficient br lr = b r /(1 ρφ) and by the dividend-growth coefficient slope downward in {b r,φ} space in just this way. The long-run return forecasting coefficients also describe a more economically interesting test region. In economic terms, we want our test region to contain draws more extreme than the observed sample. Many of the draws that produce a one-period return forecast coefficient b r larger than the sample value also have forecastable dividend growth, and dividend-yield variation is partially due to changing dividend-growth forecasts their dogs do bark; volatility tests are in them a half-success rather than the total failure they are in our data. It makes great economic sense to consider such draws closer to the null than our sample, even though the one-year return-forecast coefficient b r is greater than it is in our sample. This is how the long-run coefficients count such events, resulting in small probability values for events that really are, by this measure, more extreme than our data. The long-run return and dividend-growth forecast coefficients are also linked by an identity blr r blr d = 1, so the test is exactly the same whether one focuses on returns or dividend growth, removing the ambiguity in short-horizon coefficients. Powerful long-horizon regressions? The success of long-horizon regression tests leads us to another econometric controversy. Fama and French (1988) found that returnforecast t-statistics rise with horizon, suggesting that long-horizon return 1536

5 A Defense of Return Predictability regressions give greater statistical evidence for return forecastability. This finding has also been subject to great statistical scrutiny. Much of this literature concludes that long-horizon estimates do not, in fact, have better statistical power than one-period regressions. Boudoukh, Richardson, and Whitelaw (2006) are the most recent examples and they survey the literature. Their Table 5, top row, gives probability values for return forecasts from dividend-price ratios at 1 to 5 year horizons, based on simulations similar to mine. They report 15, 14, 13, 12 and 17% values. In short, they find no advantage to long-horizon regressions. How do I find such large power advantages for long-horizon regression coefficients? The main answer is that typical long-horizon estimates, going out to 5-year or even 10-year horizons, do not weight φ enough to see the power benefits. For example, the 2 year return coefficient = b r (1 + φ). Sinceb r 0.1, this coefficient weights variation in φ by 0.1 times as much as it weights variation in b r. But φ and b r estimates vary about one-for-one across samples, so a powerful test needs to construct a region in {b r,φ} space with about that slope, which the is b (2) r implied infinite-horizon coefficient br lr = b r /(1 ρφ) does. This finding does not imply that one should construct 30-year returns and regress them on dividend yields or other forecasting variables. I calculate long-horizon coefficients implied from the one-year regression coefficients, and they are here just a convenient way of combining those one-year regression coefficients b r,φ to generate a test region in {b r,φ} space that has good power and strong economic intuition. We therefore obtain a nice resolution of this long-running statistical controversy. I reproduce results such as Boudoukh, Richardson, and Whitelaw s (2006), that direct regressions at 1-year to 5-year horizons have little or no power advantages over 1-year regressions, but I also agree with results such as Campbell (2001) and Valkanov (2003), that there are strong power advantages to long-horizon regressions, advantages that are maximized at very long horizons and, to some extent, by calculating long-horizon statistics implied by VARs rather than direct estimates. Out-of-sample R 2 Goyal and Welch (2003, 2005) found that return forecasts based on dividend yields and a number of other variables do not work out of sample. They compared forecasts of returns at time t + 1 formed by estimating the regression using data up to time t, with forecasts that use the sample mean in the same period. They found that the sample mean produces a better out-of-sample prediction than do the return-forecasting regressions. I confirm Goyal and Welch s observation that out-of-sample return forecasts are poor, but I show that this result is to be expected. Setting up a null in which return forecasts account for all dividend-yield volatility, I find 1537

6 The Review of Financial Studies / v 21 n out-of-sample performance as bad or worse than that in the data 30 40% of the time. Thus, the Goyal Welch calculations do not provide a statistical rejection of forecastable returns. Out-of-sample R 2 is not a test; it is not a statistic that somehow gives us better power to distinguish alternatives than conventional full-sample hypothesis tests. Instead, Goyal and Welch s findings are an important caution about the practical usefulness of return forecasts in forming aggressive real-time market-timing portfolios given the persistence of forecasting variables and the short span of available data. Common misunderstandings First, one should not conclude that returns are not forecastable, but we can somehow infer their forecastability from dividend evidence. The issue is hypothesis tests, not point estimates. The point estimates are, as in Table 1, that returns are very forecastable, where the adjective very means by any economic metric. The point estimate (possibly with a bias adjustment) remains anyone s best guess. Hypothesis tests ask, What is the probability that we see something as large as Table 1 by chance, if returns are truly not forecastable? Stambaugh (1999) answer is about 15%. Even 15% is still not 50 or 90%, so zero return forecastability is still not a very likely summary of the data. Failing to reject the null does not mean that we wholeheartedly accept the i.i.d. worldview. Lots of nulls cannot be rejected. In this context, I point out that the unforecastable-return null has other implications that one can also test the implication that we should see a large dividend-growth forecast, a low dividend-yield autocorrelation, and a small long-run return forecast. Looking at these other statistics, we can say that there is in fact less than a 5% chance that our data or something more extreme is generated by a coherent world with unpredictable returns. But this evidence, like the return-based evidence, also does nothing to change the point estimate. Second, this paper is about the statistics of return forecastability, not how best to forecast returns. Simple dividend-yield regressions do not provide the strongest estimates or the best representation of return forecastability. If one really wants to forecast returns, additional variables are important, and one should pick variables and specifications that reflect repurchases, dividend smoothing, and possible changes in dividend payment behavior. I use the simplest environment in order to make the statistical points most transparently. Better specifications can only increase the evidence for forecastable returns. For this reason, the point of this article is not to vary the specification until the magic 5% barrier is crossed. The point of this article is to see how different and more comprehensive statistical analysis yields different results, and in particular how tests based on 1538

8 The Review of Financial Studies / v 21 n identity (4) implies that the regression coefficients obey the approximate identity b r = 1 ρφ + b d (5) Second, the identity (4) links the errors in (1) (3) by ε r t+1 = εd t+1 ρεdp t+1 (6) Thus, the three equations (1) (3) are redundant. One can infer the data, coefficients, and error of any one equation from those of the other two. The identity (5) shows clearly how we cannot form a null by taking b r = 0 without changing the dividend-growth forecast b d or the dividendyield autocorrelation φ. In particular, as long as φ is nonexplosive, φ<1/ρ 1.04, we cannot choose a null in which both dividend growth and returns are unforecastable b r = 0andb d = 0. To generate a coherent null with b r = 0, we must assume a negative b d, and then we must address the absence of this coefficient in the data. By subtracting inflation from both sides, Equations (4) (6) can apply to real returns and real dividend growth. Subtracting the risk-free rate from both sides, we can relate the excess log return (r t+1 r f t ) on the left-hand side of Equation (4) to dividend growth less the interest rate ( d t+1 r f t ) on the right-hand side. One can either introduce an extra interest rate term or simply understand dividend growth to include both terms. I follow the latter convention in the excess return results below. One can form similar identities and decompositions with other variables. For example, starting with the price/earnings ratio, we form a similar identity that also includes the earnings/dividend ratio. To form a null hypothesis, then, I start with estimates of Equations (1) (3) formed from regressions of log real returns, log real dividend growth and the log dividend yield in annual Center for Research in Security Prices (CRSP) data, , displayed in Table 2. The coefficients are worth keeping in mind. The return-forecasting coefficient is b r 0.10, the dividend-growth forecasting coefficient is b d 0, and the OLS estimate of the dividend-yield autocorrelation is φ The standard errors are about the same, 0.05 in each case. Alas, the identity (5) is not exact. The implied column of Table 2 gives each coefficient implied by the other two equations and the identity, in which I calculate ρ from the mean log dividend yield as ee(p d) ρ = = ee(p d) The difference is small, about in each case, but large enough to make a visible difference in the results. For example, the t-statistic calculated from the implied b r coefficient is 0.101/0.050 = 2.02 rather 1540

9 A Defense of Return Predictability Table 2 Forecasting regressions and null hypothesis ε s. d. (diagonal) Estimates and correlation. Null 1 Null 2 ˆb, ˆφ σ(ˆb) implied r d dp b,φ b,φ r d dp Each row represents an OLS forecasting regression on the log dividend yield in annual CRSP data For example, the first row presents the regression r t+1 = a r + b r (d t p t ) + ε r t+1. Standard errors σ(ˆb) include a GMM correction for heteroskedasticity. The implied column calculates each coefficient based on the other two coefficients and the identity b r = 1 ρφ + b d,usingρ = The diagonals of the ε s. d. matrix give the standard deviation of the regression errors in percent; the off-diagonals give the correlation between errors in percent. The Null columns describes coefficients used to simulate data under the null hypothesis that returns are not predictable. than 0.097/0.05 = 1.94, andwewill seeasmuchastwo to threepercentage point differences in probability values to follow. The middle three columns of Table 2 present the error standard deviations on the diagonal and correlations on the off-diagonal. Returns have almost 20% standard deviation. Dividend growth has a large 14% standard deviation. In part, this number comes from large variability in dividends in the prewar data. In part, the standard method for recovering dividends from the CRSP returns 5 means that dividends paid early in the year are reinvested at the market return to the end of the year. In part, aggregate dividends, which include all cash payouts, are in fact quite volatile. Most importantly for the joint distributions that follow, return and dividend-yield shocks are strongly negatively correlated ( 70%), in contrast to the nearly zero correlation between dividend-growth shocks and dividend-yield shocks (7.5%). The final columns of Table 2 present the coefficients of the null hypotheses I use to simulate distributions. I set b r = 0. I start by choosing φ at its sample estimate φ = I consider alternative values of φ below. 5 CRSP gives total returns R and returns without dividends Rx. I find dividend yields by I then can find dividend growth by D t+1 = R t+1 1 = P t+1 + D t+1 P t 1 P t+1 Rx t+1 P t P t+1 D t+1 = (D t+1/p t+1 ) Rx D t (D t /P t ) t+1 = D t+1 P t P t+1 P t+1 D t P t Cochrane (1991) shows that this procedure implies that dividends paid early in the year are reinvested at the return R to the end of the year. Accumulating dividends at a different rate is an attractive and frequently followed alternative, but then returns, prices, and dividends no longer obey the identity R t+1 = (P t+1 + D t+1 )/P t with end-of-year prices. 1541

10 The Review of Financial Studies / v 21 n Given b r = 0andφ, the necessary dividend forecast coefficient b d follows from the identity b d = ρφ 1 + b r 0.1. We have to choose two variables to simulate and then let the third follow from the identity (4). I simulate the dividend-growth and dividend-yield system. This is a particularly nice system, since we can interpret the errors as essentially uncorrelated shocks to expected dividend growth and shocks to actual dividend growth respectively. (Formally, the VAR (7) can be derived from a model in which expected dividend growth follows an AR(1), E t ( d t+1 ) = x t = φx t 1 + δ x t, returns are not forecastable, and dividend yields are generated from the present value identity (9)). However, the identity (4) holds well enough that this choice has almost no effect on the results. In sum, the null hypotheses takes the form d t+1 p t+1 d t+1 r t+1 = φ ρφ 1 0 (d t p t ) + ε dp t+1 ε d t+1 ε d t+1 ρεdp t+1 (7) I use the sample estimate of the covariance matrix of ε dp and ε d. I simulate 50,000 artificial data sets from each null. For φ<1, I draw the first observation d 0 p 0 from the unconditional density d 0 p 0 N [ 0,σ 2 ( ε dp) /(1 φ 2 ) ].Forφ 1, I start at d 0 p 0 = 0. I then draw ε d t and ε dp t as random normals and simulate the system forward. 2. Distribution of Regression Coefficients and t-statistics 2.1 Return and dividend-growth forecasts In each Monte Carlo draw I run regressions (1) (3). Figure 1 plots the joint distribution of the return b r and dividend-growth b d coefficients, and the joint distribution of their t-statistics. Table 3 collects probabilities. The marginal distribution of the return-forecast coefficient b r gives quite weak evidence against the unforecastable-return null. The Monte Carlo draw produces a coefficient larger than the sample estimate 22% of the time, and a larger t-statistic than the sample about 10% of the time (points to the right of the vertical line in the top panels of Figure 1, top left entries of Table 3). Taken on its own, we cannot reject the hypothesis that the return-forecasting coefficient b r is zero at the conventional 5% level. This finding confirms the results of Goetzmann and Jorion (1993), Nelson and Kim (1993), and Stambaugh (1999). However, the null must assume that dividend growth is forecastable. As a result, almost all simulations give a large negative dividend-growth forecast coefficient b d. The null and cloud of estimates in Figure 1 are vertically centered a good deal below zero and below the horizontal line of the sample estimate ˆb d. Dividend-growth forecasting coefficients larger 1542

11 A Defense of Return Predictability Table 3 Percent probability values under the φ = null b r t r b d t d Real Excess Each column gives the probability that the indicated coefficients are greater than their sample values, under the null. Monte Carlo simulation of the null described in Table 2 with 50,000 draws. b d Coefficients, φ = % 1.8 % % % b r t, b d t-stats, φ = % 1.6 % 8.7 % 89.7 % t, b r b d Coefficients, φ = % 5.6 % % % b r t, b d t-stats, φ = % 5.7 % 11.1 % 82.4 % t, b r Figure 1 Joint distribution of return and dividend-growth forecasting coefficients (left) and t-statistics (right). The lines and large dots give the sample estimates. The triangle gives the null. One thousand simulations are plotted for clarity; each point represents 1/10% probability. Percentages are the fraction of 50,000 simulations that fall in the indicated quadrants. than the roughly zero values observed in sample are seen only 1.77% of the time, and the dividend-growth t-statistic is only greater than its roughly zero sample value 1.67% of the time (points above the horizontal lines in Figure 1, b d and t d columns of Table 3). Results are even stronger for excess returns, for which b d > ˆb d is observed only 1.11% of the time and the t-statistic only 0.87% of the time (Table 3). In sum, the lack of dividend forecastability in the data gives far stronger statistical evidence against the null than does the presence of return 1543

14 The Review of Financial Studies / v 21 n In fact, the point estimates in Table 4 show slightly more than 100% of dividend-yield volatility coming from returns, since the point estimate of dividend-growth forecasts go slightly the wrong way. The decomposition (10) is not a decomposition into orthogonal components, so elements can be greater than 100% or less than 0% in this way. Excess returns in the last row of Table 4 show slightly stronger results. In the point estimates, high price-dividend ratios actually signal slightly higher interest rates, so they signal even lower excess returns. The first two rows of Table 4 drive home the fact that, by the identity br lr bd lr = 1, the long-horizon dividend-growth regression gives exactly the same results as the long-horizon return regression. 6 The standard errors are also exactly the same, and the t-statistic for br lr = 0isexactlythe same as the t statistic for bd lr = 1. One great advantage of using long-horizon regression coefficients is that we do not need to choose between return and dividend-growth tests, as they give precisely the same results. As a result, we can tabulate the small-sample distribution of the test in a conventional histogram, rather than a two-dimensional plot. Figure 2 tabulates the small-sample distribution of the long-run returnforecast coefficients, and Table 4 includes the probability values how many long-run return forecasts are greater than the sample value under the unforecastable-return null blr r = 0. There is about a 1.5% probability value of seeing a long-run forecast larger than seen in the data. (The range of probability values in Table 4 derives from the fact that the identities are only approximate, so the result depends on which of the three parameters (b r,φ,b d ) is implied from the other two.) The long-run return (or dividend-growth) regressions give essentially the same strong rejections as the short-run dividend-growth regression. The last row of Table 4 shows the results for excess returns. Again, excess returns paint a stronger picture. The probability values of % are lower and the evidence against the null even stronger. 3. Power, Correlation, and the φ View Where does the greater power of dividend-growth and long-run return tests come from? How do we relate these results to the usual analysis of the {b r,φ} coefficients in a two-variable VAR consisting of the return and the forecasting variable? 6 The identities are only approximate, so to display estimates that obey the identities one must estimate two of b r, b d,andφ, and imply the other using the identity b r b d = 1 ρφ. Inthetoptwolinesof Table 4, I use the direct ˆb r and ˆb d estimates from Table 2. I then use ρ ˆφ impl = 1 ˆb r + ˆb d and I construct long-run estimates by ˆb r lr = ˆb r /(1 ρ ˆφ impl ).Since ˆφ = 0.94 and ˆφ impl = 0.95, the difference between these estimates and those that use ˆφ is very small. Using the direct estimate ˆφ rather than ˆφ impl, we have ˆb lr r = 1.04 (s.e. = 0.42) and b d lr = 0.08 (s.e. = 0.42). 1546

15 A Defense of Return Predictability φ = 0.94 φ = 0.99 Data Data b r /(1-ρφ) b r /(1-ρφ) Figure 2 Distribution of br lr = b r /(1 ρφ). The vertical bar gives the corresponding value in the data. b r and φ, φ = 0.94 b r and φ, φ = % 0.0 % % 0.6 % % % φ % lr b r b d φ % lr b r b d b r b r Figure 3 Joint distribution of return and dividend yield forecasting coefficients b r,φ. In each graph the triangle marks the null hypothesis used to generate the data and the circle marks the estimated coefficients ˆb r, ˆφ. The diagonal dashed line marked b d marks the line b r = 1 ρφ + ˆb d ; points above and to the right are draws in which b d exceeds its sample value. The solid diagonal line marked br lr marks the line defined by b r /(1 ρφ) = ˆb r /(1 ρ ˆφ); points above and to the right are draws in which br lr exceeds its sample value. Numbers are the percentage of the draws that fall in the indicated quadrants. Figure 3 addresses these questions by plotting the joint distribution of estimates {b r,φ} across simulations. We see again that a high return coefficient b r by itself is not so unusual, occurring about 22% of the time (area to the right of the vertical line). We learn, however, that b r and φ estimates are negatively correlated across samples. Though we often see large b r and large φ, we almost never see b r larger than in the data together with φ larger than in the data. (Figure 3 is the same as Lewellen (2004) Figure 1, Panel B, except Lewellen calibrates to monthly postwar data. Lewellen focuses on a different distributional calculation.) This observation on its own is not a good way to form a test statistic. Though the northeast quadrant of the plot is suggestively empty, we would 1547

16 The Review of Financial Studies / v 21 n not want to commit to accepting the null for φ just below a rectangular rejection region and arbitrarily large b r. The {b r,φ} plot is more important to help us to digest why the dividendgrowth test b d and the long-horizon regression test br lr give so many fewer rejections under then null than the usual one-period return b r test. The diagonal dashed line marked b d in Figure 3 uses the identity b r = 1 ρφ + b d to mark the region b d > ˆb d in this {b r,φ} space. Points above and to the right of this dashed line are exactly the points above b d > ˆb d in Figure 1. The similar diagonal solid line marked br lr uses the definition b lr = b r / (1 ρφ) to mark the region b lr > ˆb lr in this {b r,φ} r space. Points above and to the right of this line are exactly the points above br lr > ˆb r lr in the histogram of Figure 2. Viewed in {b r,φ} space, dividend-growth b d and long-run regression b lr tests capture in a single number and in a sensible test region the fact that samples with high b r typically come with low φ, and they exploit that negative correlation to produce more powerful tests. By the definition b r / (1 ρφ), samples with high b r but low φ produce a low long-run return forecast br lr. Generating a large long-run return forecast requires both a large one-year return forecast and a large autocorrelation, so forecasts can build with horizon. Because most large return forecasts come with low autocorrelation, it is much harder for the null to deliver a large long-run return forecast. By the identity b d = b r + ρφ 1, dividend growth works the same way. r r 3.1 The source of negative correlation The strong negative correlation of estimates b r and φ across samples, which underlies the power of long-horizon return and dividend-growth tests, stems from the strong negative correlation of the shocks ε r t+1 and εdp t+1 in the underlying VAR, (1) (3). If shocks are negatively correlated in two regressions with the same right-hand variable, then a draw of shocks that produces an unusually large coefficient in the first regression corresponds to a draw of shocks that produces an unusually small coefficient in the second regression. It is important to understand this correlation. We do not want the power of long-run or dividend-growth tests to hinge on some arbitrary and inessential feature of the null hypothesis, and strong correlations of shocks are usually not central parts of a specification. From the identity ε r t+1 = εd t+1 ρεdp t+1 (13) the fact that return shocks and dividend-yield shocks are strongly and negatively correlated is equivalent to the fact that dividend-yield shocks and dividend-growth shocks are not correlated. Intuitively, we can see this 1548

17 A Defense of Return Predictability fact by looking at the definition of return R t+1 = (1 + P t+1/d t+1 ) P t /D t D t+1 D t that underlies (13): a decline in dividend yield D t+1 /P t+1 is a rise in prices, which raises returns, but only so long as there is no offsetting change in dividend growth D t+1 /D t. More precisely, multiply both sides of (13) by ε dp t+1 and take expectations, yielding ( ) cov ε r t+1,εdp t+1 = cov ( ε dp t+1,εd t+1 ) ( ) ρσ 2 ε dp t+1 (14) When dividend growth and dividend yields are uncorrelated cov(ε dp t+1,εd t+1 ) = 0, we obtain a strong negative correlation between returns and dividend yields cov(ε r t+1,εdp t+1 ). Dividend yields move on news of expected returns (in the point estimates) or news of expected dividend growth (in the null) (see (9)). Thus, the central fact in our data is that shocks to expected returns (data) or expected dividend growth (null) are uncorrelated with shocks to ex post dividend growth. The strong negative correlation of dividend-yield shocks with return shocks follows from the definition of a return. It seems that we can easily imagine other structures, however. For example, in typical time-series processes, like an AR(1), shocks to ex post dividend growth are correlated with shocks to expected dividend growth; only rather special cases do not display this correlation. In economic models, it is not inconceivable that a negative shock to current dividends would raise risk premia, raising expected returns and thus dividend yields. However, identity (13) makes it hard to construct plausible alternatives. There are only three degrees of freedom in the variance-covariance matrix of the three shocks, since any one variable can be completely determined from the other two. As a result, changing one correlation forces us to change the rest of the covariance matrix in deeply counterfactual ways. For example, let us try to construct a covariance matrix in which return and dividend-yield shocks are uncorrelated, cov(ε r t+1,εdp t+1 ) = 0. Let us continue to match the volatility of returns σ(ε r t+1 ) = 0.2 and dividend yields σ(ε dp t+1 ) = We cannot, however, match the volatility of dividend growth. Writing the identity (13) as ε d t+1 = εr t+1 + ρεdp t+1 (15) we see that in order to produce cov(ε r t+1,εdp t+1 ) = 0, we must specify dividend-growth shocks that are more volatile than returns! We need to specify 25% dividend-growth volatility, rather than the 14% volatility in 1549

18 The Review of Financial Studies / v 21 n our data: σ(ε d ) = σ 2 (ε r t+1 ) + ρ2 σ 2 (ε dp t+1 ) = = 0.25 It is a quite robust fact that return variation is dominated by variation of prices or valuations with little change in cashflows, and more so at high frequencies. The variance of returns far exceeds the variance of dividend growth. By (15) that fact alone implies that positive innovations to current returns ε r t+1 must come with negative innovations to dividend yields. Continuing the example, we can find the required correlation of dividend-growth shocks and dividend-yield shocks by multiplying (15) by ε dp t+1 giving cov ( ε d,ε dp ) = cov(ε r,ε dp ) + ρσ 2 ( ε dp ) corr ( ε d,ε dp) = ρ σ ( ε dp) σ(ε d ) = = 0.58 rather than 0.07, essentially zero, in the data (Table 2). In this alternative world, good news about dividend growth is frequently accompanied by an increase in dividend yield, meaning prices do not move that much. In turn, that means the good news about dividend growth comes either with news that future dividend growth will be low dividends have a large mean-reverting component or with news that future expected returns will be high. In our data, dividend-yield shocks typically raise prices proportionally, leading to no correlation with dividend yields. Needless to say, the changes to the covariance matrix required to generate a positive correlation between return and dividend-yield shocks are even more extreme. In sum, the negative correlation of estimates {b r,φ}, which ultimately derives from the negative correlation of shocks ε r t+1,εdp t+1 or equivalently from the near-zero correlation of shocks ε dp t+1,εd t+1 is a deep and essential feature of the data, not an easily changed auxiliary to the null. 3.2 Which is the right region? economics We now have three tests: the one-period regression coefficients b r and b d, and the long-horizon regression coefficient b lr. Which is the right one to look at? Should we test b r > ˆb r, or should we test b d > ˆb d, or br lr > ˆb r lr? Or perhaps we should test some other subset of the {b r,φ} region? The central underlying question is, how should we form a single test statistic from the joint distribution of many parameters? We have three parameters, b r,b d,φ. The identity b r = 1 ρφ + b d means we can reduce the issue to a two-dimensional space, but we still have two dimensions to think about. 1550

19 A Defense of Return Predictability In economic terms, we want the most interesting test. The issue comes down to defining what is the event we have seen, and what other events we would consider more extreme, or further from the null than the event we have seen. If we focus on the one-year return regression, we think of the event as the return forecast coefficient seen in the data b r = ˆb r 0.1, and more extreme events as those with greater oneyear return-forecast coefficients, b r > ˆb r. But, as the joint distributions point out, most of the events with b r > ˆb r, have dividend-growth forecast coefficients larger (more negative) than seen in the data, b d < ˆb d,they have dividend-yield autocorrelations lower than seen in the data φ<ˆφ, they have long-run return coefficients less than seen in the data br lr < ˆb r lr, and thus (by identity) they have long-run dividend-growth coefficients larger (more negative) than seen in the data, bd lr < ˆb d lr 0. In these events, dividend-growth is forecastable, prices are moving to some extent on forecasts of future dividend growth, and in the right direction. Volatility tests are a half-success, rather than the total failure that they are in our data. The long-run coefficients count these draws as closer to the null than our data, despite the larger values of b r. From this point of view, the test on the long-run coefficient is the economically interesting test. If we want to view that test in the {b r,φ} space of one-period regression coefficients, diagonal test regions as marked by br lr in Figure 3 are the right ones to look at. The dividend-growth coefficient tests b d > ˆb d give almost exactly the same answers as the long-run coefficient tests, as can be seen both in the tables and by the fact that the dividend b d and long-run br lr regions of Figure 3 are nearly the same. In fact, these tests are different conceptually and slightly different in this sample. The long-run return coefficient test br lr > ˆb r lr means bd lr > ˆb d lr which means b d/(1 ρφ) > ˆb d /(1 ρ ˆφ). Ifwe had ˆb d = 0 exactly, this would mean b d > ˆb d = 0 and the two regions would be exactly the same. With ˆb d 0, a different sample φ can affect the long-run dividend-growth coefficient bd lr = b d/(1 ρφ) for a given value of b d, perhaps pushing it across a boundary. In a sample with ˆb d further from zero, the two test statistics could give substantially different answers. When there is a difference, I find the long-run coefficients more economically attractive than the dividend-growth coefficients. As an important practical example, think about the specification of the null. In long-run coefficient terms, we specify the null as bd lr = 1,blr r = 0, that is, all variation in dividend yields is due to time-varying expected dividend growth and none to time-varying expected returns. In short-run coefficient terms, this specification is equivalent to b d = 1/(1 ρφ). At the sample φ = ˆφ 0.96, we have b d 0.1. As we vary φ, however, we vary b d to keep bd lr = 1. This is exactly how I specify the φ = 0.99 null above and how I specify the null for different φ values below. 1551

20 The Review of Financial Studies / v 21 n Suppose instead that we specify the null in short-run coefficient terms as b d = 0.1 for any value of φ. Now, different values of φ give us specifications in which expected returns do explain nonzero fractions of dividend yield and in which returns are predictable. For example, at φ = 0.99, we would have b lr = 0.1/( ) 2 and thus br lr d 1, with b r ( ) ( 1) In this null, a rise in prices signals so much higher dividend growth that it must also signal much higher future returns. Obviously, this is not a very interesting way to express the null hypothesis that returns are unpredictable. The same sorts of things happen to test regions if b d 0. If one accepts that the null should be expressed this way, in terms of long-horizon coefficients to accommodate variation in φ, it seems almost inescapable that the economically interesting test region should be specified in the same way. 3.3 Which is the right region? statistics In statistical terms we want the most powerful test. It is clear that the dividend growth b d and long-run br lr tests, implying a test of a diagonal region in {b r,φ} space, are more powerful. It is important to understand the source of that power. Power, of course, is not the probability under the null of finding more extreme statistics that I have calculated. To document power, I should set up regions based on b r, b d,andb lr that reject at a given level, say 5% of the time, under the null. Then I should evaluate the probability that draws enter those rejection regions under alternatives, in particular generating data from the estimated parameters ˆb r and ˆφ. I should document that draws do enter the long-run or dividend-growth rejection regions more frequently. I do not perform these calculations in the interest of brevity, since it is clear from the graphs how they work out. Since the b r vertical line in Figure 3 demarks a 22% probability value now, the boundary of the 5% region under the null is farther to the right. Since the b d and b lr diagonal lines demark 1 2% probability values now, the boundaries of the 5% regions under the null are a bit to the left of the current lines. Under the alternative, the cloud of points in Figure 3 moves to the right drag the triangle to the circle and move the cloud of points with it. Because of the negative correlation between b r and φ estimates, that operation will drag roughly half of the simulated data points across the diagonal lines, but it will still leave the bulk of the data points shy of the 5% vertical b r region. The long-run and dividend-growth tests do have more power. Therefore, one key to the extra power is the negative correlation between b r and φ coefficients, documented by Figure 3. If the cloud sloped the other way, there would be no power advantage. The other key to extra power is a limitation on φ in the null. So far, I have calculated test statistics from a point null, specifying {b r = 0,φ = 0.941}. 1552

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